# Local Finiteness of the Twisted Bruhat Orders on Affine Weyl Groups

## TL;DR

This paper proves local finiteness of twisted strong Bruhat orders on affine Weyl groups and explores their properties, including infinite antichains and applications to oriented matroids.

## Contribution

It establishes the local finiteness of twisted strong Bruhat orders on affine Weyl groups, extending previous results, and investigates their structural properties and applications.

## Key findings

- Twisted strong Bruhat order on affine Weyl groups is locally finite.
- Existence of infinite antichains in twisted Bruhat orders for certain sets.
- Application of twisted weak Bruhat order to oriented matroids.

## Abstract

In this paper, we investigate various properties of strong and weak twisted Bruhat orders on a Coxeter group. In particular, we prove that any twisted strong Bruhat order on an affine Weyl group is locally finite, strengthening a result of Dyer in J. Algebra, 163, 861--879 (1994). We also show that for a non-finite and non-cofinite biclosed set $B$ in the positive system of an affine root system with rank greater than 2, the set of elements having a fixed $B$-twisted length is infinite. This implies that the twisted strong and weak Bruhat orders have an infinite antichain in those cases. Finally, we show that twisted weak Bruhat order can be applied to the study of the tope poset of an infinite oriented matroid arising from an affine root system.